Functional Limit Theorem for the Empirical Process of a Class of Bernoulli Shifts with Long Memory
Résumé
We prove a functional central limit theorem for the empirical process of a stationary process $X_t = Y_t + V_t$, where $Y_t$ is a long memory moving average in i.i.d. r.v.'s $\zeta_s, s\le t $, and $V_t = V(\zeta_t, \zeta_{t-1}, \dots )$ is a weakly dependent nonlinear Bernoulli shift. Conditions of weak dependence of $V_t$ are written in terms of $L^2-$norms of shift-cut differences $ V(\zeta_t, \dots, \zeta_{t-n}, 0, \dots, ) - V(\zeta_t, \dots, \zeta_{t-n+1}, 0, \dots )$. Examples of Bernoulli shifts are discussed. The limit empirical process is a degenerated process of the form $f(x) Z $, where $f$ is the marginal p.d.f. of $X_0$ and $Z $ is a standard normal r.v. The proof is based on a uniform reduction principle for the empirical process.
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